I must admit that it’s the first time that I’ve heard of such rule for forming possessives in English. What I know is this general rule: To form the possessive of a singular noun that doesn’t end in the letter “s,” simply affix an apostrophe-s (’s) to that noun, as in “Lea’s blouse” and “Rigoberto’s excuse.” However, if the noun is plural or already ends in “-s”, just affix an apostrophe (’), as in “the mayors’ entourage” and “the winners’ prize” (for plural nouns) and “Ces’ TV show” and “the hippopotamus’ snout” (for singular nouns ending in “-s”). I’m not aware of any rule that makes a noun’s syllable count as a determinant for the use of either the apostrophe-s or just an apostrophe.
There are several special cases that deviate from this general rule, though. For the possessive of names ending in “-s”, it’s possible for the written form to either affix apostrophe-s to the noun, as in “Jones’s dilemma” and “Harris’s itinerary,” or just an apostrophe, as in “Jones’ dilemma” and “Harris’ itinerary.” It’s a stylistic choice that once made, needs to be used consistently. We also must keep in mind that for the spoken form in both cases, the /z/ sound must be added to the name to evoke the plural.
Applying this rule to the examples of possessives that you provided, we have the following stylistic choices: “Lucas’ dog” and “Kris’ son” (using apostrophe only), or “Lucas’s dog” and “Kris’s son” (using apostrophe-s). We actually have the same stylistic choices for proper names with any number of syllables, such as “Mercedes’ gown” and “Amaryllis’ hairstyle” (using apostrophe only) or “Mercedes’s gown” and “Amaryllis’s hairstyle” (using apostrophe-s). As you can see, the same rule for making the possessive applies regardless of the number of syllables in the noun.
For a more comprehensive discussion of forming possessives in English, particularly with respect to special cases, click this link to the
CCC Foundation Grammar Guide to Writing. I’m sure the site can clarify for you many aspects of this simple but sometimes tricky aspect of English grammar.
